Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bending of Light around a Star

  1. Mar 8, 2006 #1
    I know that the formula for the angle of bending of light around the star [itex]\theta \;[/itex] in radians is:

    [itex]\theta \; = \frac{4GM}{rc^2}[/itex]

    This is equal to full bending of light, half of which occurs on the rear side of the star and the other half occuring on the foreground side.

    So half of this is equal to:

    [itex]\frac{\theta \;}{2} = \frac{2GM}{rc^2}[/itex]

    So that:

    [itex]dt = \frac{d\tau \;}{\sqrt{1-\frac{\theta\;}{2}}}[/itex]

    [itex]\sqrt{1-\frac{\theta \;}{2}} = \frac{d\tau\;}{dt}[/itex]

    [itex]1-\frac{\theta \;}{2} = \frac{d\tau\;^2}{dt^2}[/itex]

    [itex]1-\frac{d\tau \;^2}{dt^2} = \frac{\theta\;}{2}[/itex]

    [itex]2 \left( 1-\frac{d\tau \;^2}{dt^2} \right) = \theta\;[/itex]

    If [itex]\theta \;=2 radians[/itex] then there would be infinite gravitational time dilation.

    Did I get that right?
     
    Last edited: Mar 8, 2006
  2. jcsd
  3. Mar 8, 2006 #2

    pervect

    User Avatar
    Staff Emeritus
    Science Advisor

    I'm not sure about your notation. Also, the formula is only valid for small angles.

    My text gives the angle of deflection (converting to non-geometric units) as

    [tex]\Theta = \frac{4GM}{c^2b}[/tex]

    where b is the impact parameter, related to the Schwarzschild radius of the 'turning point" [itex]r_{tp}[/itex] by

    [tex]
    b = \frac{r_{tp}}{\sqrt{1-\frac{2GM}{r_{tp}c^2}}}
    [/tex]

    This is only a small-angle approximation, the results won't be right for [itex]\Theta = 2\pi[/itex].

    A photon that gets deflected by a black hole will always be outside the photon sphere at r=3GM/c^2, so the maximum time dilation at closest approach for the actual orbit will always be finite - less than [itex]\sqrt{3}[/itex].

    [add]I've attached a numeric solution for a photon starting out just outside the photon radius at r_min=3.01, spiraling outwards. The mass M of the black hole is 1 in geometric units.

    The appropriate diffeq used to generate it which describes the exact orbit of a photon around a Schwarzschild black hole is , using geometric units in which G=c=1:

    [tex]
    \left( \frac{1}{r^2} \frac{d r}{d \theta} \right) ^2 + \frac{1-2M/r}{r^2} = \frac{1}{b^2} = \frac{1-2M/r_{min}}{r_{min}^2}
    [/tex]
     

    Attached Files:

    Last edited: Mar 8, 2006
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Bending Light around Date
I Bending of light - caused by gravity or relativity? Aug 31, 2017
A Gravitational Bending of Light? Aug 23, 2017
A Bending of light formula Jul 26, 2017
B The bending of light on a single point Feb 27, 2017
B The moon and light that bends around it Nov 9, 2016